Singular points of real quartic curves via computer algebra

TL;DR

This paper uses computer algebra to classify all singular points of real and complex quartic curves, providing a complete proof that leverages symbolic computations of Puiseux expansions.

Contribution

It offers a complete, computer-assisted classification of singular points for real and complex quartic curves, clarifying the computational process involved.

Findings

Thirteen types of singular points for irreducible real quartic curves.

Seventeen types of singular points for reducible real quartic curves.

Complete classification with proof using Maple.

Abstract

There are thirteen types of singular points for irreducible real quartic curves and seventeen types of singular points for reducible real quartic curves. This classification is originally due to D.A. Gudkov. There are nine types of singular points for irreducible complex quartic curves and ten types of singular points for reducible complex quartic curves. We derive the complete classification with proof by using the computer algebra system Maple. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. Thus, the proof consists of a sequence of large symbolic computations that can be done nicely using Maple.